Let
$G$
be a finite group scheme operating on an algebraic variety
$X$
, both defined over an algebraically closed field
$k$
. The paper first investigates the properties of the quotient morphism
$X\longrightarrow X/G$
over the open subset of
$X$
consisting of points whose stabilizers have maximal index in
$G$
. Given a
$G$
-linearized coherent sheaf on
$X$
, it describes similarly an open subset of
$X$
over which the invariants in the sheaf behave nicely in some way. The points in
$X$
with linearly reductive stabilizers are characterized in representation theoretic terms. It is shown that the set of such points is nonempty if and only if the field of rational functions
$k(X)$
is an injective
$G$
-module. Applications of these results to the invariants of a restricted Lie algebra
${\frak g}$
operating on the function ring
$k[X]$
by derivations are considered in the final section. Furthermore, conditions are found ensuring that the ring
$k[X]^{\frak g}$
is generated over the subring of
$p$
th powers in
$k[X]$
, where
$p={\rm char}\,k>0$
, by a given system of invariant functions and is a locally complete intersection.